Coclass theory for nilpotent semigroups via their associated algebras

Coclass theory has been a highly successful approach towards the investigation and classification of finite nilpotent groups. Here we suggest a variation of this approach for finite nilpotent semigroups: we propose to study coclass graphs for the contracted semigroup algebras of nilpotent semigroups. We exhibit a series of conjectures on the shape of these coclass graphs. If these are proven, then this reduces the classification of nilpotent semigroups of a fixed coclass to a finite calculation. We show that our conjectures are supported by the nilpotent semigroups of coclass 0 and 1. Computational experiments suggest that the conjectures also hold for the nilpotent semigroups of coclass 2 and 3.